I agree that you are in a tough situation. I can't "make it all better", but I can respond to some of your issues:
Dreamhost is Located in Southern California, and it is the very beginning of a Monday business day there. Given the fact that Dreamhost currently hosts over 300,000 domains, I suspect the phones are very busy right now. The lack of answer is probably because the phone is on a rotary, and is trying to find an available line. I suggest you either set the fax machine on autodial and let it keep trying till it gets through, or try again later. Given the international nature of Dreamhost's business, I could only guess when the fax/phone might be more likely answered
It's always best to keep the old host until the new one is set up, but there is little you can do about that now. On the "up" side, as long as your domain at the old registrar was not "locked" and they did receive the transfer request, the "newish" ICANN rules no longer require an affirmative response from you to allow the transfer to take place...if there is no response instructing them to stop the transfer within 5 days, the transfer request is supposed to be honored. If you have the ability to telephone the registrar, you could attempt to clarify the situation with them that way, and maybe get them to expedite (though that is not too likely given they are supposed to allow 5 days for the email contact to object to the transfer).
As for how long you should expect for this to take, given the issues you have, and depending upon when you get the fax issue it's hard to tell. Once the fax issue is resolved, Dreamhost should, and does, set you up immediately. As per issue 2 above, you could be looking at 5-7 days for the transfer to be complete (though it is often less), which is what Dreamhost tells you when you initiate the transfer.
No. The transfer should proceed as long as you don't object by responding to that email telling them not to transfer the domain.
I understand your frustration, but there is nothing else you can do except send a message detailing you situation to Dreamhost via the Contact Form on the Main Dreamhost Website and keep trying to send the fax. If you already have a Web Id to log into the panel, you can also generate a support ticket from the Control Panel-->Support-->Contact Support link. That said, I'd keep trying the fax, as early this morning there were over 1,000 tech support requests in the queue.
There is a fax machine there; you just have not been able to get it to answer (it is no doubt "busy", but you probably will not hear the signal because of the rotary management of the lines). There is no voice line for Dreamhost that is answered.
For what it is worth, you are not the only one having difficulty with this.
Not much help, I'm sure, but it is the best info I can give you. Good luck with it all; I hope you can get it sorted and will enjoy the hosting here.